Nonlocal Differential Model Research Articles

Bending and buckling analysis of functionally graded Timoshenko nanobeam using Two-Phase Local/Nonlocal piezoelectric integral model

Previous studies indicate that nonlocal piezoelectric differential model would lead to an inconsistent bending response of Euler-Bernoulli nanobeam. In this paper, static bending and elastic buckling of functionally graded piezoelectric (FGP) Timoshenko nanobeam are studied through two-phase local/nonlocal piezoelectric integral model. The differential governing equations and standard boundary conditions are derived on the basis of the principle of minimum potential energy. The relations between general strain and nonlocal stress components are expressed as integral forms and then transformed equivalently into differential forms with constitutive boundary conditions. Several nominal variables are introduced to simplify the mathematical formulation. It shows that Timoshenko nanobeam based on purely nonlocal piezoelectric integral model would lead to ill-posed mathematical model and inconsistent size-dependent response with and without considering constitutive boundary conditions, respectively. The general differential quadrature method is applied to obtain the numerical results for bending deflections and buckling loads. The influence of nonlocal parameters, boundary conditions, functionally graded index and buckling order on the bending deflections and buckling loads is investigated numerically. A consistent softening response is obtained.

Structural analysis of nonlocal nanobeam via FEM using equivalent nonlocal differential model

Structural analysis of nonlocal nanobeam via FEM using equivalent nonlocal differential model

On the stress analysis of anisotropic curved panels

Stress generally occurs in curved panels made of non-isotropic materials (triclinic materials with 21 in-dependent elastic components) in nature. Hence, the behavior of triclinic shells remains poorly understood. The present study aims to obtain a model with no approximation in processing to truly analyze the static bending characteristics of nano-size shells made of triclinic material; different curves are also examined. The problem of different boundary conditions such as simply-supported, clamped, and a combination of them will also be examined. Initially, we model the panel with a higher-order shear deformation theory where Eringen nonlocal differential model is selected to predict the size-dependency. Then, the governing motion equations are acquired through a virtual work of Hamilton statement. Ultimately, the partial differential equations for static bending problems are solved numerically utilizing the generalized differential quadrature method. The numerical examples are expanded for the sensitivity of stress and displacements’ deflection to geometrical coefficients, boundary conditions, nonlocality and curves. Furthermore, a comparison between the present anisotropic model and the isotropic approximated matrix of elastic components is performed to show the importance of the present work. The results presented here can not solely be a computational study of triclinic shells but can also be used as a benchmark of future works on materials with unsymmetrical crystalline structures.

On the consistency of two-phase local/nonlocal piezoelectric integral model

In this paper, we propose general strain- and stress-driven two-phase local/nonlocal piezoelectric integral models, which can distinguish the difference of nonlocal effects on the elastic and piezoelectric behaviors of nanostructures. The nonlocal piezoelectric model is transformed from integral to an equivalent differential form with four constitutive boundary conditions due to the difficulty in solving intergro-differential equations directly. The nonlocal piezoelectric integral models are used to model the static bending of the Euler-Bernoulli piezoelectric beam on the assumption that the nonlocal elastic and piezoelectric parameters are coincident with each other. The governing differential equations as well as constitutive and standard boundary conditions are deduced. It is found that purely strain- and stress-driven nonlocal piezoelectric integral models are ill-posed, because the total number of differential orders for governing equations is less than that of boundary conditions. Meanwhile, the traditional nonlocal piezoelectric differential model would lead to inconsistent bending response for Euler-Bernoulli piezoelectric beam under different boundary and loading conditions. Several nominal variables are introduced to normalize the governing equations and boundary conditions, and the general differential quadrature method (GDQM) is used to obtain the numerical solutions. The results from current models are validated against results in the literature. It is clearly established that a consistent softening and toughening effects can be obtained for static bending of the Euler-Bernoulli beam based on the general strain- and stress-driven local/nonlocal piezoelectric integral models, respectively.

AĞIRLIKLI ARTIKLAR KULLANILARAK NANOÇUBUKLARIN EKSENEL STATİK ANALİZİ İÇİN KESİN ÇÖZÜMLER

In the present work, axial static analysis of nanorods under triangular loading is presented via Eringen’s nonlocal differential model. Three weighted residual methods (Subdomain, Galerkin and Least squares methods) are used to obtain the exact static deflection. These methods require that the integral of the error with different assumptions over the domain be set to zero. The number of equations have to be equal to unknown terms. A cubic displacement function has been chosen for three weighted residual methods. Subdomain, Galerkin and Least squares methods yield identical solution as the exact solution. The plots of the solution are shown for different number of unknown coefficients.

Postbuckling analysis of nonlocal functionally graded beams

Abstract The main goal of this research is to study the postbuckling behavior of nonlocal functionally graded beams. Eringen’s nonlocal differential model is used to evaluate the influence of the material length scale in the bending response. An improved shear deformation beam theory with five independent parameters is utilized, which is suitable for the use of 3D constitutive equations. A finite element model is derived with spectral high-order interpolation functions to avoid shear locking. The formulation is verified by comparing the present results with the ones found in the literature. Functionally graded beams with different boundary conditions, nonlocal parameters, and power law indices are analyzed. It is shown that the present model can accurately predict the behavior of nonlocal beams due to the use of high-order terms in the displacement field in comparison with classical beam formulations. Finally, new benchmark problems are analyzed to show the capabilities of the present model to evaluate the effect of the nonlocal parameter and the power law index on postbuckling beam behavior.

Studying free hight-frequency vibrations of an inhomogeneous nanorod based on the nonlocal theory of elasticity

Free high-frequency longitudinal vibrations of an inhomogeneous nanosized rod are studied on the basis of the nonlocal theory of elasticity. The upper part of spectrum with the wavelength comparable to the internal characteristic dimension of a nanorod is examined. An equations in the integral form with the Helmholtz kernel, incorporating both local and nonlocal phases, is used as the constitutive one. The original integro-differential equation is reduced to the forth-order differential equation with variable coefficients, the pair of additional boundary conditions being deduced. UsingWKB-method, a solution of the boundaryvalue problem is constructed in the form of the superposition of a main solution and edge effect integrals. As an alternative model, we consider the purely nonlocal (one-phase) differential model which allows estimating the upper part of spectrum of eigen-frequencies. Considering the nanorod with a variable cross-section area, we revealed a fair convergence of eigen-frequencies found in the framework of two models when the local fraction in the two-phase model vanishes.

A rational beam-elastic substrate model with incorporation of beam-bulk nonlocality and surface-free energy

In this study, a rational beam-elastic substrate model with inclusion of nonlocal and surface-energy effects is developed for bending and buckling analyses of nanobeams lying on elastic substrate media. The beam-section kinematics follows Euler–Bernoulli beam hypothesis. The thermodynamics-based strain gradient theory is employed to represent the beam-bulk nonlocality while the Gurtin–Murdoch surface theory is utilized to account for the surface-free energy. Interaction between the beam and its underlying substrate medium is described by the Winkler foundation model. The governing equilibrium equation and admissible natural boundary conditions are consistently obtained using the virtual displacement principle. To characterize bending and buckling responses of the new beam-elastic substrate model, three numerical examples are employed: the first demonstrates the capability of the proposed model in eliminating the paradoxical behavior present in the Eringen nonlocal differential model; the second characterizes the bending response of the free–free nanobeam-elastic substrate system; and the third examines the influences of several model parameters as well as the size-dependent effect on the buckling response of the simply supported nanobeam-elastic substrate system.

Dynamics of nonlocal thick nano-bars

The thick bar model, accounting for the lateral deformation, shear stiffness, and lateral inertia effect, is the most comprehensive structural theory to study the axial deformation of carbon nanotubes. Physically motivated definition of the axial force field and associated higher order boundary conditions are determined applying a consistent variational framework. The effects of long-range interactions are suitably realized in the framework of the nonlocal integral elasticity. The integral convolutions of the nonlocal constitutive law are determined and suitably resorted with the equivalent nonlocal differential model equipped with non-standard boundary conditions. Preceding contributions on the elastodynamic analysis of the elastic thick bar are, therefore, amended by properly taking into account the higher order and non-standard boundary conditions. The established size-dependent thick bar model is demonstrated to be exempt from the inherent drawbacks of the nonlocal differential formulation and leads to well-posed elastodynamic problems. The wave desperation response and free vibrational behavior of elastic thick bars with kinematic constraints of nano-mechanics interest are rigorously investigated by making recourse to a viable solution approach. New numerical benchmarks are detected for the elastodynamic response of nonlocal thick nano-bars. A consistent approach for nanoscopic study of the field quantities in the nonlocal mechanics is proposed that is capable of properly confirming the smaller-is-softer phenomenon.

Free vibration and buckling analysis of elastically supported transversely inhomogeneous functionally graded nanoplate in thermal environment using Rayleigh–Ritz method

In this study, free vibration and buckling behaviors of a functionally graded nanoplate supported by the Winkler–Pasternak foundation using a nonlocal classical plate theory are investigated. Eringen’s nonlocal differential model has been used for considering the small-scale effect. The properties of the functionally graded nanoplate are considered to vary transversely following the power law. The governing vibration and buckling equations of an elastically supported functionally graded nanoplate have been derived using the principle of virtual work, and the solution is obtained using the Rayleigh–Ritz method and characteristic polynomials. The advantage of this method is that it disposes of all the drawbacks regarding edge constraints. The objective of the article is to see the effect of edge constraints, aspect ratios, material property exponent, nonlocal parameter, and foundation parameters on the nondimensionalized frequency and the buckling load of an embedded functionally graded nanoplate in a thermal environment. The study highlights that the nonlocal effect is pronounced for higher modes and/or higher aspect ratios and need to be considered for the analysis of the nanoplate. Further, it is observed that the effect of the Pasternak foundation is prominent on nondimensionalized frequencies and buckling of the functionally graded nanoplate.

Shear-horizontal waves in periodic layered nanostructure with both nonlocal and interface effects

The propagation of shear-horizontal (SH) waves in the periodic layered nanocomposite is investigated by using both the nonlocal integral model and the nonlocal differential model with the interface effect. Based on the transfer matrix method and the Bloch theory, the band structures for SH waves with both vertical and oblique incidences to the structure are obtained. It is found that by choosing appropriate interface parameters, the dispersion curves predicted by the nonlocal differential model with the interface effect can be tuned to be the same as those based on the nonlocal integral model. Thus, by propagating the SH waves vertically and obliquely to the periodic layered nanostructure, we could invert, respectively, the interface mass density and the interface shear modulus, by matching the dispersion curves. Examples are further shown on how to determine the interface mass density and the interface shear modulus in theory.

Well-posed nonlocal elasticity model for finite domains and its application to the mechanical behavior of nanorods

Eringen’s nonlocal elasticity theory is one of the most attractive approaches to investigate the intrinsic scale effect of nanoscopic structures. Eringen proposed both integral and differential nonlocal models which are equivalent to each other over unbounded continuous domains. Although the Eringen nonlocal models can be used as very useful tools for modeling the mechanical characteristics of nanoscopic structures, however, several researchers have reported some paradoxical results when they used the nonlocal differential model. In this paper, we develop a well-posed nonlocal differential model for finite domains, and its applicability to predict the static and dynamic behavior of a nanorod is investigated. It is shown that the proposed integral and differential nonlocal models are equivalent to each other over bounded continuous domains, and the corresponding elastic problems are well-posed and consistent. In addition, some paradigmatic static problems are solved the and we show that the paradoxical results disappear by using the present model.

A Thermodynamics-Based Nonlocal Bar-Elastic Substrate Model with Inclusion of Surface-Energy Effect

This paper presents a bar-elastic substrate model to investigate the axial responses of nanowire-elastic substrate systems considering the effects of nonlocality and surface energy. The thermodynamics-based strain gradient model is adopted to capture the nonlocality of the bar-bulk material while the Gurtin-Murdoch surface theory is utilized to consider the surface energy. To characterize the bar-surrounding substrate interaction, the Winkler foundation model is employed. In a direct manner, system compatibility conditions are obtained while within the framework of the virtual displacement principle, the system equilibrium condition and the corresponding natural boundary conditions are consistently obtained. Three numerical simulations are conducted to investigate the characteristics and behaviors of the nanowire-elastic substrate system: the first is conducted to reveal the capability of the proposed model to eliminate the paradoxical behavior inherent to the Eringen nonlocal differential model; the second is employed to characterize responses of the nanowire-elastic substrate system; and the third is aimed at demonstrating the dependence of the system effective Young’s modulus on several system parameters.

Stress-driven nonlocal elasticity for the instability analysis of fluid-conveying C-BN hybrid-nanotube in a magneto-thermal environment

The Eringen’s strain-driven nonlocal differential model is well-established to exhibit inconsistencies when applied to bounded continua of applicative interest. The stress-driven nonlocal theory leads instead to well-posed nonlocal elastic formulations demonstrating stiffening structural responses. In the present article, using the stress-driven nonlocal model, a comprehensive analysis is conducted to explore the vibrational characteristics and critical divergence velocity of a hybrid-nanotube constructed by carbon (C) and boron nitride (BN) nanotubes conveying magnetic fluid. The impact of size-dependence, magnetic field and thermal medium on the dynamic behavior of the systems is included in the proposed model. The obtained governing equations of two-segment nanotubes are then examined using the finite element method. It is interestingly showed that the threshold of the divergence/flutter instability of the system would be enhanced by employing a hetero-nanotube instead of a nanotube composed of a uniform material. Furthermore, the results demonstrate that the configuration of the mode shapes may be dramatically changed for a nanotube conveying fluid. Therefore, the classical modes do no longer exist, and should not be considered in the dynamics of the system. It is also shown that by assuming a low temperature medium, the critical velocity increases by increasing the temperature and decreases in the case of high temperature.

Torsional dynamic response of viscoelastic SWCNT subjected to linear and harmonic torques with general boundary conditions via Eringen’s nonlocal differential model

The present work investigates the dynamic torsional vibrations in a single-walled carbon nanotube embedded in a viscoelastic medium under the linear and harmonic external torques. In order to derive the equation of motion and associated clamped–clamped and clamped–torsional spring boundary conditions, Hamilton’s principle is utilized. Eringen’s nonlocal elasticity theory is the selected theory to show the small-scale effect. The assumed modes method and a Galerkin method are applied to analyze the derived equation analytically as the exact solutions. Then, the Rayleigh–Ritz method is proposed to compare the obtained results. In free vibration analysis, the nonlocal parameter has an effect on both natural frequencies and energy dissipation in different modes. It should be emphasized that the forced analysis of torsional vibration in the viscoelastic nanoscale is novel. The variation of the nonlocal parameter, thickness of carbon nanotube, damping coefficient, stiffness of the boundary spring, as well as the influence of excitation frequency on the angular displacement in the time domain, are illustrated in this paper. The results desirably are consistent with those from other studies.

Eigenfrequencies of microtubules embedded in the cytoplasm by means of the nonlocal integral elasticity

A biologically microscopic system presenting a highly scientific interest is the microtubule (MT). Our research endeavor revolves around the eigenfrequencies’ analysis of an MT embedded in the cytoplasm of the cell by means of the nonlocal integral elasticity for the first time. The MT is simulated as a beam and the cytoplasm as a Pasternak-type elastic foundation, respectively. The responses of the nonlocal integral stress models show to have a softening behavior in comparison with that of the classic model. Unlike the nonlocal differential model, no paradoxes and inconsistencies are raised for the nonlocal integral models. Our research conclusions are a hopeful sign for the applications of biomaterials and bioengineering structures.

Bending Analysis of Nonlocal Functionally Graded Beams

In this paper, we study the nonlocal linear bending behavior of functionally graded beams subjected to distributed loads. A finite element formulation for an improved first-order shear deformation theory for beams with five independent variables is proposed. The formulation takes into consideration 3D constitutive equations. Eringen’s nonlocal differential model is used to rewrite the nonlocal stress resultants in terms of displacements. The finite element formulation is derived by means of the principle of virtual work. High-order nodal-spectral interpolation functions were utilized to approximate the field variables, which minimizes the locking problem. Numerical results and comparisons of the present formulation with those found in the literature for typical benchmark problems involving nonlocal beams are found to be satisfactory and show the validity of the developed finite element model.

Nonlocal Adhesion Models for Microorganisms on Bounded Domains

In 2006 Armstrong, Painter, and Sherratt formulated a nonlocal differential equation model for cell-cell adhesion. For the one-dimensional case on a bounded domain we derive various types of biological boundary conditions, describing adhesive, repulsive, and neutral boundaries. We prove local and global existence and uniqueness for the resulting integrodifferential equations. In numerical simulations we consider adhesive, repulsive, and neutral boundary conditions, and we show that the solutions mimic known behavior of fluid adhesion to boundaries. In addition, we observe interior pattern formation due to cell-cell adhesion.

Exact and asymptotic bending analysis of microbeams under different boundary conditions using stress‐derived nonlocal integral model

AbstractEringen's nonlocal differential elasticity is widely applied to model nano‐ and micro‐structures, although previous studies have shown some inconsistences, e.g., the nonlocal parameter maybe has no effect on the deflection of Euler‐Bernoulli and Timoshenko beams subjected to some kinds of boundary and loading conditions. In this paper, the static bending analysis of Euler‐Bernoulli and Timoshenko beams is performed with the application of stress‐driven nonlocal integral model. The Fredholm type integral constitutive equations of the first kind are transformed to Volterra integral equations of the first kind through simply adjusting the limit of integrals, and the general solutions to the deflection as well as bending moment and so on are derived and obtained explicitly through solving the integro‐differential governing equations with the Laplace transformation. Exact solutions are derived explicitly for different loading and boundary conditions, especially for the paradoxical beam problem while using nonlocal differential model. The analytical and asymptotic expressions of the beam deflections obtained in this paper are validated against to those existing analytical and numerical results in literature. It is clearly established that, with the application of the stress‐driven nonlocal integral model, a consistent toughening effect can be obtained for both Euler‐Bernoulli and Timoshenko beams.

Nonlinear transverse vibration of nano-strings based on the differential type of nonlocal theory

The nonlinear vibration responses of nano-strings are studied based on the theory of Eringen’s nonlocal elasticity. Firstly, the nonlocal differential constitutive model in one-dimensional form which is suitable for a string structure is used, and then the governing equation of motion for the nonlinear vibration of nano-strings is derived by considering the expression of classical Lagrangian strain. In order to solve the non-dimensional nonlinear governing equation of motion, the Galerkin method or Rayleigh-Ritz method is applied, and the nonlinear partial differential equation is approximately transformed into the a set of ordinary differential equations. The ordinary differential equations are then solved by a numerical method, and the nonlinear vibration responses under different time histories are thus obtained. Subsequently, the approximate numerical solution of the nonlinear displacement is solved by the second-order multi-scale method. The nonlinear phenomena in the transverse displacement and the influences of nonlocal scale parameter on the nonlinear vibration characteristics of nano-strings are analyzed accordingly. The results will provide a basis for understanding and controlling the nonlinear dynamics of nano-strings which may act as key components in the booming intelligent nano-systems.